{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T20:19:48Z","timestamp":1782418788491,"version":"3.54.5"},"reference-count":108,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T00:00:00Z","timestamp":1627689600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T00:00:00Z","timestamp":1627689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"name":"INBRE-NM","award":["P20GM103451"],"award-info":[{"award-number":["P20GM103451"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s40314-021-01592-6","type":"journal-article","created":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T18:02:48Z","timestamp":1627754568000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Qualitative analysis of a mathematical model with presymptomatic individuals and two SARS-CoV-2 variants"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5847-678X","authenticated-orcid":false,"given":"Gilberto","family":"Gonz\u00e1lez-Parra","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abraham J.","family":"Arenas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,7,31]]},"reference":[{"key":"1592_CR1","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/j.ijid.2020.10.091","volume":"102","author":"M Al-Qahtani","year":"2020","unstructured":"Al-Qahtani M, AlAli S, AbdulRahman A, Alsayyad AS, Otoom S, Atkin SL (2020) The prevalence of asymptomatic and symptomatic COVID-19 in a cohort of quarantined subjects. Int J Infect Dis 102:285\u2013288","journal-title":"Int J Infect Dis"},{"issue":"6534","key":"1592_CR2","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1126\/science.abg7404","volume":"371","author":"DM Altmann","year":"2021","unstructured":"Altmann DM, Boyton RJ, Beale R (2021) Immunity to SARS-CoV-2 variants of concern. Science 371(6534):1103\u20131104","journal-title":"Science"},{"key":"1592_CR3","doi-asserted-by":"crossref","unstructured":"Alves THE, de Souza TA, de Almeida Silva S, Ramos NA, de Oliveira SV (2020) Underreporting of death by COVID-19 in Brazil\u2019s second most populous state. Front Public Health 8","DOI":"10.3389\/fpubh.2020.578645"},{"issue":"4","key":"1592_CR4","doi-asserted-by":"publisher","first-page":"1992","DOI":"10.3934\/mbe.2019098","volume":"16","author":"J Amador Pacheco","year":"2019","unstructured":"Amador Pacheco J, Armesto D, G\u00f3mez-Corral A (2019) Extreme values in sir epidemic models with two strains and cross-immunity. Math Biosci Eng 16(4):1992\u20132022","journal-title":"Math Biosci Eng"},{"issue":"2","key":"1592_CR5","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1086\/524065","volume":"197","author":"V Andreasen","year":"2008","unstructured":"Andreasen V, Viboud C, Simonsen L (2008) Epidemiologic characterization of the $$1918$$ influenza pandemic summer wave in Copenhagen: implications for pandemic control strategies. J Infect Dis 197(2):270\u2013278","journal-title":"J Infect Dis"},{"key":"1592_CR6","doi-asserted-by":"crossref","unstructured":"Arvisais-Anhalt S, Lehmann CU, Park JY, Araj E, Holcomb M, Jamieson AR, McDonald S, Medford RJ, Perl TM, Toomay SM et al (2020) What the coronavirus disease 2019 (COVID-19) pandemic has reinforced: the need for accurate data. Clin Infect Dis","DOI":"10.1093\/cid\/ciaa1686"},{"issue":"7","key":"1592_CR7","doi-asserted-by":"publisher","first-page":"1176","DOI":"10.1002\/sim.6015","volume":"33","author":"A Azmon","year":"2014","unstructured":"Azmon A, Faes C, Hens N (2014) On the estimation of the reproduction number based on misreported epidemic data. Stat Med 33(7):1176\u20131192","journal-title":"Stat Med"},{"issue":"14","key":"1592_CR8","doi-asserted-by":"publisher","first-page":"1406","DOI":"10.1001\/jama.2020.2565","volume":"323","author":"Y Bai","year":"2020","unstructured":"Bai Y, Yao L, Wei T, Tian F, Jin DY, Chen L, Wang M (2020) Presumed asymptomatic carrier transmission of COVID-19. JAMA 323(14):1406\u20131407","journal-title":"JAMA"},{"issue":"2","key":"1592_CR9","doi-asserted-by":"publisher","first-page":"78","DOI":"10.3390\/vaccines9020078","volume":"9","author":"J Benest","year":"2021","unstructured":"Benest J, Rhodes S, Quaife M, Evans TG, White RG (2021) Optimising vaccine dose in inoculation against SARS-CoV-2, a multi-factor optimisation modelling study to maximise vaccine safety and efficacy. Vaccines 9(2):78","journal-title":"Vaccines"},{"issue":"02","key":"1592_CR10","doi-asserted-by":"publisher","first-page":"1950021","DOI":"10.1142\/S1793524519500219","volume":"12","author":"D Bentaleb","year":"2019","unstructured":"Bentaleb D, Amine S (2019) Lyapunov function and global stability for a two-strain seir model with bilinear and non-monotone incidence. Int J Biomath 12(02):1950021","journal-title":"Int J Biomath"},{"issue":"1","key":"1592_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/1471-2334-14-116","volume":"14","author":"H Bernard","year":"2014","unstructured":"Bernard H, Werber D, H\u00f6hle M (2014) Estimating the under-reporting of norovirus illness in Germany utilizing enhanced awareness of diarrhoea during a large outbreak of Shiga toxin-producing E. coli o104: H4 in 2011-a time series analysis. BMC Infect Dis 14(1):1\u20136","journal-title":"BMC Infect Dis"},{"issue":"2","key":"1592_CR12","first-page":"113","volume":"2","author":"F Brauer","year":"2017","unstructured":"Brauer F (2017) Mathematical epidemiology: Past, present, and future. Infect Dis Modell 2(2):113\u2013127","journal-title":"Infect Dis Modell"},{"key":"1592_CR13","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3516-1","volume-title":"Mathematical models in population biology and epidemiology","author":"F Brauer","year":"2001","unstructured":"Brauer F, Castillo-Chavez C, Castillo-Chavez C (2001) Mathematical models in population biology and epidemiology, vol 40. Springer, New York"},{"issue":"9","key":"1592_CR14","doi-asserted-by":"publisher","first-page":"e1003346","DOI":"10.1371\/journal.pmed.1003346","volume":"17","author":"D Buitrago-Garcia","year":"2020","unstructured":"Buitrago-Garcia D, Egli-Gany D, Counotte MJ, Hossmann S, Imeri H, Ipekci AM, Salanti G, Low N (2020) Occurrence and transmission potential of asymptomatic and presymptomatic sars-cov-2 infections: a living systematic review and meta-analysis. PLoS Med 17(9):e1003346","journal-title":"PLoS Med"},{"issue":"5","key":"1592_CR15","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1016\/S1473-3099(20)30303-0","volume":"20","author":"T Burki","year":"2020","unstructured":"Burki T (2020) COVID-19 in Latin America. Lancet Infect Dis 20(5):547\u2013548","journal-title":"Lancet Infect Dis"},{"issue":"4","key":"1592_CR16","first-page":"223","volume":"5","author":"O Byambasuren","year":"2020","unstructured":"Byambasuren O, Cardona M, Bell K, Clark J, McLaws ML, Glasziou P (2020) Estimating the extent of asymptomatic COVID-19 and its potential for community transmission: systematic review and meta-analysis. Off J Assoc Med Microbiol Infect Dis Canada 5(4):223\u2013234","journal-title":"Off J Assoc Med Microbiol Infect Dis Canada"},{"key":"1592_CR17","unstructured":"Centers for Disease Control and Prevention (2020) https:\/\/www.cdc.gov\/coronavirus\/2019-nCoV\/index.html"},{"issue":"9","key":"1592_CR18","doi-asserted-by":"publisher","first-page":"1969","DOI":"10.1681\/ASN.2020060827","volume":"31","author":"C Clarke","year":"2020","unstructured":"Clarke C, Prendecki M, Dhutia A, Ali MA, Sajjad H, Shivakumar O, Lightstone L, Kelleher P, Pickering MC, Thomas D et al (2020) High prevalence of asymptomatic COVID-19 infection in hemodialysis patients detected using serologic screening. J Am Soc Nephrol 31(9):1969\u20131975","journal-title":"J Am Soc Nephrol"},{"key":"1592_CR19","doi-asserted-by":"crossref","unstructured":"Davies NG, Abbott S, Barnard RC, Jarvis CI, Kucharski AJ, Munday JD, Pearson CA, Russell TW, Tully DC, Washburne AD et al (2021) Estimated transmissibility and impact of SARS-CoV-2 lineage B. 1.1. 7 in England. Science","DOI":"10.1101\/2020.12.24.20248822"},{"issue":"12","key":"1592_CR20","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","volume":"180","author":"PV den Driessche","year":"2002","unstructured":"den Driessche PV, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(12):29\u201348","journal-title":"Math Biosci"},{"key":"1592_CR21","doi-asserted-by":"crossref","unstructured":"Dinleyici EC, Borrow R, Safadi MAP, van Damme P, Munoz FM (2020) Vaccines and routine immunization strategies during the COVID-19 pandemic. Human vaccines & immunotherapeutics 1\u20138","DOI":"10.1080\/21645515.2020.1804776"},{"issue":"2","key":"1592_CR22","doi-asserted-by":"publisher","first-page":"224","DOI":"10.5935\/0103-507X.20200030","volume":"32","author":"MF Do Prado","year":"2020","unstructured":"Do Prado MF, de Paula Antunes BB, Bastos LDSL, Peres IT, Da Silva ADAB, Dantas LF, Bai\u00e3o FA, Ma\u00e7aira P, Hamacher S, Bozza FA (2020) Analysis of COVID-19 under-reporting in Brazil. Revista Brasileira de terapia intensiva 32(2):224","journal-title":"Revista Brasileira de terapia intensiva"},{"issue":"8","key":"1592_CR23","doi-asserted-by":"publisher","first-page":"e0236976","DOI":"10.1371\/journal.pone.0236976","volume":"15","author":"HM Dobrovolny","year":"2020","unstructured":"Dobrovolny HM (2020) Modeling the role of asymptomatics in infection spread with application to sars-cov-2. PLoS ONE 15(8):e0236976","journal-title":"PLoS ONE"},{"key":"1592_CR24","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.virol.2020.07.015","volume":"550","author":"HM Dobrovolny","year":"2020","unstructured":"Dobrovolny HM (2020) Quantifying the effect of remdesivir in rhesus macaques infected with SARS-CoV-2. Virology 550:61\u201369","journal-title":"Virology"},{"key":"1592_CR25","doi-asserted-by":"crossref","unstructured":"Dom\u00e9nech-Carb\u00f3 A, Dom\u00e9nech-Casas\u00fas C (2021) The evolution of COVID-19: a discontinuous approach. Physica A: Stat Mech Appl p 125752","DOI":"10.1016\/j.physa.2021.125752"},{"key":"1592_CR26","unstructured":"Faria NR, Mellan TA, Whittaker C, Claro IM, Candido DdS, Mishra S, Crispim MA, Sales FC, Hawryluk I, McCrone JT, et al (2021) Genomics and epidemiology of the P. 1 SARS-CoV-2 lineage in Manaus, Brazil. Science"},{"key":"1592_CR27","doi-asserted-by":"publisher","unstructured":"Ferguson NM, Laydon D, Nedjati-Gilani G, Imai N, Ainslie K, Baguelin M, Bhatia S, Boonyasiri A, Cucunub\u00e1 Z, Cuomo-Dannenburg G et al (2020) Impact of non-pharmaceutical interventions (npis) to reduce COVID-19 mortality and healthcare demand. Imperial College, London. https:\/\/doi.org\/10.25561\/77482","DOI":"10.25561\/77482"},{"key":"1592_CR28","doi-asserted-by":"crossref","unstructured":"Fiorentini S, Messali S, Zani A, Caccuri F, Giovanetti M, Ciccozzi M, Caruso A (2021) First detection of SARS-CoV-2 spike protein N501 mutation in Italy in August, 2020. The Lancet Infectious Diseases","DOI":"10.1016\/S1473-3099(21)00007-4"},{"key":"1592_CR29","unstructured":"Fred Brauer JAN (1989) The Qualitative Theory of Ordinary Differential Equations: An Introduction. Dover Publications,"},{"issue":"3","key":"1592_CR30","doi-asserted-by":"publisher","first-page":"95","DOI":"10.15585\/mmwr.mm7003e2","volume":"70","author":"SE Galloway","year":"2021","unstructured":"Galloway SE, Paul P, MacCannell DR, Johansson MA, Brooks JT, MacNeil A, Slayton RB, Tong S, Silk BJ, Armstrong GL et al (2021) Emergence of SARS-CoV-2 b. 1. 1. 7 lineage-united states, december 29, 2020-january 12, 2021. Morb Mortal Wkly Rep 70(3):95","journal-title":"Morb Mortal Wkly Rep"},{"key":"1592_CR31","doi-asserted-by":"publisher","unstructured":"Gandhi M, Yokoe DS, Havlir DV (2020) Asymptomatic transmission, the achilles\u2019 heel of current strategies to control Covid-19. New England Journal of Medicine 382(22):2158\u20132160. https:\/\/doi.org\/10.1056\/NEJMe2009758","DOI":"10.1056\/NEJMe2009758"},{"issue":"1","key":"1592_CR32","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1002\/num.21804","volume":"30","author":"G Gonz\u00e1lez-Parra","year":"2014","unstructured":"Gonz\u00e1lez-Parra G, Arenas AJ, Chen-Charpentier BM (2014) Positive numerical solution for a nonarbitrage liquidity model using nonstandard finite difference schemes. Numer Methods Part Differ Equ 30(1):210\u2013221","journal-title":"Numer Methods Part Differ Equ"},{"key":"1592_CR33","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez-Parra G, Arenas A, Aranda Diego F, LS (2011) Modeling the epidemic waves of AH1N1\/09 influenza around the world. Spatial Spatio-temp Epidemiol 2:219\u2013226","DOI":"10.1016\/j.sste.2011.05.002"},{"issue":"4","key":"1592_CR34","doi-asserted-by":"publisher","first-page":"794","DOI":"10.1016\/j.cell.2020.06.040","volume":"182","author":"ND Grubaugh","year":"2020","unstructured":"Grubaugh ND, Hanage WP, Rasmussen AL (2020) Making sense of mutation: what D614G means for the COVID-19 pandemic remains unclear. Cell 182(4):794\u2013795","journal-title":"Cell"},{"key":"1592_CR35","doi-asserted-by":"crossref","unstructured":"Gupta RK (2021) Will SARS-CoV-2 variants of concern affect the promise of vaccines? Nat Rev Immunol 1\u20132","DOI":"10.1038\/s41577-021-00556-5"},{"issue":"4","key":"1592_CR36","doi-asserted-by":"publisher","first-page":"739","DOI":"10.3390\/vaccines8040739","volume":"8","author":"A Haque","year":"2020","unstructured":"Haque A, Pant AB (2020) Efforts at COVID-19 vaccine development: challenges and successes. Vaccines 8(4):739","journal-title":"Vaccines"},{"key":"1592_CR37","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1137\/S0036144500371907","volume":"42","author":"HW Hethcote","year":"2005","unstructured":"Hethcote HW (2005) Mathematics of infectious diseases. SIAM Rev 42:599\u2013653","journal-title":"SIAM Rev"},{"key":"1592_CR38","doi-asserted-by":"crossref","unstructured":"Hoseinpour Dehkordi A, Alizadeh M, Derakhshan P, Babazadeh P, Jahandideh A (2020) Understanding epidemic data and statistics: A case study of COVID-19. Journal of medical virology","DOI":"10.1101\/2020.03.15.20036418"},{"key":"1592_CR39","doi-asserted-by":"publisher","first-page":"n230","DOI":"10.1136\/bmj.n230","volume":"372","author":"G Iacobucci","year":"2021","unstructured":"Iacobucci G (2021) Covid-19: new UK variant may be linked to increased death rate, early data indicate. BMJ 372:n230","journal-title":"BMJ"},{"key":"1592_CR40","unstructured":"IHME COVID-19 Forecasting Team: Modeling COVID-19 scenarios for the United States. Nat Med (2020)"},{"key":"1592_CR41","doi-asserted-by":"publisher","first-page":"105303","DOI":"10.1016\/j.cnsns.2020.105303","volume":"88","author":"B Ivorra","year":"2020","unstructured":"Ivorra B, Ferr\u00e1ndez MR, Vela-P\u00e9rez M, Ramos A (2020) Mathematical modeling of the spread of the coronavirus disease 2019 (covid-19) taking into account the undetected infections. The case of China. Commun Nonlinear Sci Numer Simul 88:105303","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"1","key":"1592_CR42","doi-asserted-by":"publisher","first-page":"e2035057","DOI":"10.1001\/jamanetworkopen.2020.35057","volume":"4","author":"MA Johansson","year":"2021","unstructured":"Johansson MA, Quandelacy TM, Kada S, Prasad PV, Steele M, Brooks JT, Slayton RB, Biggerstaff M, Butler JC (2021) SARS-CoV-2 transmission from people without COVID-19 symptoms. JAMA Network Open 4(1):e2035057\u2013e2035057","journal-title":"JAMA Network Open"},{"key":"1592_CR43","unstructured":"Johns Hopkins University and Medicine: https:\/\/coronavirus.jhu.edu (2020)"},{"key":"1592_CR44","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/j.jtbi.2016.09.025","volume":"412","author":"S Kim","year":"2017","unstructured":"Kim S, Lee J, Jung E (2017) Mathematical model of transmission dynamics and optimal control strategies for 2009 A\/H1N1 influenza in the Republic of Korea. J Theor Biol 412:74\u201385","journal-title":"J Theor Biol"},{"issue":"6","key":"1592_CR45","doi-asserted-by":"publisher","first-page":"e0232580","DOI":"10.1371\/journal.pone.0232580","volume":"15","author":"Y Kim","year":"2020","unstructured":"Kim Y, Barber AV, Lee S (2020) Modeling influenza transmission dynamics with media coverage data of the 2009 H1N1 outbreak in Korea. PLoS ONE 15(6):e0232580","journal-title":"PLoS ONE"},{"key":"1592_CR46","first-page":"43","volume":"35","author":"A Koirala","year":"2020","unstructured":"Koirala A, Joo YJ, Khatami A, Chiu C, Britton PN (2020) Vaccines for COVID-19: the current state of play. Paediatr Respir Rev 35:43\u201349","journal-title":"Paediatr Respir Rev"},{"issue":"1","key":"1592_CR47","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1080\/17513758.2021.1883130","volume":"15","author":"JD Kong","year":"2021","unstructured":"Kong JD, Tchuendom RF, Adeleye SA, David JF, Admasu FS, Bakare EA, Siewe N (2021) SARS-CoV-2 and self-medication in Cameroon: a mathematical model. J Biol Dyn 15(1):137\u2013150","journal-title":"J Biol Dyn"},{"issue":"4","key":"1592_CR48","doi-asserted-by":"publisher","first-page":"812","DOI":"10.1016\/j.cell.2020.06.043","volume":"182","author":"B Korber","year":"2020","unstructured":"Korber B, Fischer WM, Gnanakaran S, Yoon H, Theiler J, Abfalterer W, Hengartner N, Giorgi EE, Bhattacharya T, Foley B et al (2020) Tracking changes in SARS-CoV-2 spike: evidence that D614G increases infectivity of the COVID-19 virus. Cell 182(4):812\u2013827","journal-title":"Cell"},{"key":"1592_CR49","doi-asserted-by":"crossref","unstructured":"Kucharski AJ, Russell TW, Diamond C, Liu Y, Edmunds J, Funk S, Eggo RM, Sun F, Jit M, Munday JD et al (2020) Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The Lancet infectious diseases","DOI":"10.1101\/2020.01.31.20019901"},{"issue":"3","key":"1592_CR50","doi-asserted-by":"publisher","first-page":"789","DOI":"10.3390\/jcm9030789","volume":"9","author":"T Kuniya","year":"2020","unstructured":"Kuniya T (2020) Prediction of the epidemic peak of coronavirus disease in Japan, 2020. J Clin Med 9(3):789","journal-title":"J Clin Med"},{"issue":"6528","key":"1592_CR51","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1126\/science.371.6528.448","volume":"371","author":"K Kupferschmidt","year":"2021","unstructured":"Kupferschmidt K (2021) Vaccinemakers ponder how to adapt to virus variants. Science (New York, NY) 371(6528):448\u2013449","journal-title":"Science (New York, NY)"},{"key":"1592_CR52","volume-title":"Computational methods in ordinary differential equations","author":"JD Lambert","year":"1973","unstructured":"Lambert JD (1973) Computational methods in ordinary differential equations. Wiley, New York"},{"key":"1592_CR53","doi-asserted-by":"crossref","unstructured":"Lau H, Khosrawipour T, Kocbach P, Ichii H, Bania J, Khosrawipour V (2020) Evaluating the massive underreporting and undertesting of COVID-19 cases in multiple global epicenters. Pulmonology","DOI":"10.1016\/j.pulmoe.2020.05.015"},{"key":"1592_CR54","doi-asserted-by":"publisher","unstructured":"Le Page M (2021) Threats from new variants. New Sci 249(3316):8\u20139. https:\/\/doi.org\/10.1016\/S0262-4079(21)00003-8. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0262407921000038","DOI":"10.1016\/S0262-4079(21)00003-8"},{"issue":"4","key":"1592_CR55","doi-asserted-by":"publisher","first-page":"610","DOI":"10.1017\/S0950268806007217","volume":"135","author":"J Legrand","year":"2007","unstructured":"Legrand J, Grais RF, Boelle PY, Valleron AJ, Flahault A (2007) Understanding the dynamics of ebola epidemics. Epidemiol Infect 135(4):610\u2013621","journal-title":"Epidemiol Infect"},{"key":"1592_CR56","unstructured":"Lemieux JE, Li JZ (2021) Uncovering ways that emerging SARS-CoV-2 lineages may increase transmissibility. The Journal of Infectious Diseases"},{"issue":"1","key":"1592_CR57","doi-asserted-by":"publisher","first-page":"2002106","DOI":"10.2807\/1560-7917.ES.2020.26.1.2002106","volume":"26","author":"K Leung","year":"2021","unstructured":"Leung K, Shum MH, Leung GM, Lam TT, Wu JT (2021) Early transmissibility assessment of the N501Y mutant strains of SARS-CoV-2 in the United Kingdom, October to November 2020. Eurosurveillance 26(1):2002106","journal-title":"Eurosurveillance"},{"issue":"5","key":"1592_CR58","doi-asserted-by":"publisher","first-page":"1284","DOI":"10.1016\/j.cell.2020.07.012","volume":"182","author":"Q Li","year":"2020","unstructured":"Li Q, Wu J, Nie J, Zhang L, Hao H, Liu S, Zhao C, Zhang Q, Liu H, Nie L et al (2020) The impact of mutations in SARS-CoV-2 spike on viral infectivity and antigenicity. Cell 182(5):1284\u20131294","journal-title":"Cell"},{"key":"1592_CR59","doi-asserted-by":"crossref","unstructured":"Li Q, Guan X, Wu P, Wang X, Zhou L, Tong Y, Ren R, Leung KS, Lau EH, Wong JY et al (2020) Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia. New Engl J Med","DOI":"10.1056\/NEJMoa2001316"},{"issue":"21","key":"1592_CR60","doi-asserted-by":"publisher","first-page":"1969","DOI":"10.1056\/NEJMp2005630","volume":"382","author":"N Lurie","year":"2020","unstructured":"Lurie N, Saville M, Hatchett R, Halton J (2020) Developing Covid-19 vaccines at pandemic speed. N Engl J Med 382(21):1969\u20131973","journal-title":"N Engl J Med"},{"issue":"2\u20133","key":"1592_CR61","first-page":"190","volume":"151","author":"S Mandal","year":"2020","unstructured":"Mandal S, Bhatnagar T, Arinaminpathy N, Agarwal A, Chowdhury A, Murhekar M, Gangakhedkar RR, Sarkar S (2020) Prudent public health intervention strategies to control the coronavirus disease 2019 transmission in India: a mathematical model-based approach. Indian J Med Res 151(2\u20133):190","journal-title":"Indian J Med Res"},{"issue":"2","key":"1592_CR62","doi-asserted-by":"publisher","first-page":"140","DOI":"10.3390\/epidemiologia2020012","volume":"2","author":"D Martinez-Rodriguez","year":"2021","unstructured":"Martinez-Rodriguez D, Gonzalez-Parra G, Villanueva RJ (2021) Analysis of key factors of a SARS-CoV-2 vaccination program: a mathematical modeling approach. Epidemiologia 2(2):140\u2013161","journal-title":"Epidemiologia"},{"key":"1592_CR63","doi-asserted-by":"crossref","unstructured":"Mbogo RW, Orwa TO (2021) SARS-COV-2 outbreak and control in Kenya-Mathematical model analysis. Infect Dis Model","DOI":"10.1016\/j.idm.2021.01.009"},{"key":"1592_CR64","doi-asserted-by":"publisher","first-page":"109647109647","DOI":"10.1016\/j.chaos.2020.109647","volume":"133","author":"A Meskaf","year":"2020","unstructured":"Meskaf A, Khyar O, Danane J, Allali K (2020) Global stability analysis of a two-strain epidemic model with non-monotone incidence rates. Chaos Solit Fract 133:109647109647","journal-title":"Chaos Solit Fract"},{"key":"1592_CR65","doi-asserted-by":"publisher","DOI":"10.1142\/5884","volume-title":"Advances in the applications of nonstandard finite difference schemes","author":"RE Mickens","year":"2005","unstructured":"Mickens RE (2005) Advances in the applications of nonstandard finite difference schemes. World Scientific, Singapore"},{"issue":"1","key":"1592_CR66","doi-asserted-by":"publisher","first-page":"748","DOI":"10.1080\/17513758.2020.1823494","volume":"14","author":"ARJ Mumbu","year":"2020","unstructured":"Mumbu ARJ, Hugo AK (2020) Mathematical modelling on COVID-19 transmission impacts with preventive measures: a case study of Tanzania. J Biol Dyn 14(1):748\u2013766","journal-title":"J Biol Dyn"},{"key":"1592_CR67","unstructured":"Murray JD (2002) Mathematical biology I: an introduction, Vol. 17 of interdisciplinary applied mathematics. Springer, New York, NY, USA"},{"key":"1592_CR68","doi-asserted-by":"crossref","unstructured":"Oran DP, Topol EJ (2020) Prevalence of asymptomatic SARS-CoV-2 infection: a narrative review. Ann Internal Med","DOI":"10.7326\/M20-3012"},{"key":"1592_CR69","doi-asserted-by":"crossref","unstructured":"O\u2019Regan SM, Kelly TC, Korobeinikov A, O\u2019Callaghan MJ, Pokrovskii AV (2010) Lyapunov functions for SIR and SIRS epidemic models. Appl Math Lett 23(4):446\u2013448","DOI":"10.1016\/j.aml.2009.11.014"},{"key":"1592_CR70","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s12967-020-02344-6","volume":"18","author":"M Pachetti","year":"2020","unstructured":"Pachetti M, Marini B, Benedetti F, Giudici F, Mauro E, Storici P, Masciovecchio C, Angeletti S, Ciccozzi M, Gallo RC et al (2020) Emerging SARS-CoV-2 mutation hot spots include a novel RNA-dependent-RNA polymerase variant. J Transl Med 18:1\u20139","journal-title":"J Transl Med"},{"issue":"1","key":"1592_CR71","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s12967-020-02501-x","volume":"18","author":"M Pachetti","year":"2020","unstructured":"Pachetti M, Marini B, Giudici F, Benedetti F, Angeletti S, Ciccozzi M, Masciovecchio C, Ippodrino R, Zella D (2020) Impact of lockdown on Covid-19 case fatality rate and viral mutations spread in 7 countries in Europe and North America. J Transl Med 18(1):1\u20137","journal-title":"J Transl Med"},{"key":"1592_CR72","doi-asserted-by":"crossref","unstructured":"Paltiel AD, Schwartz JL, Zheng A, Walensky RP (2020) Clinical outcomes of a COVID-19 vaccine: Implementation over efficacy: Study examines how definitions and thresholds of vaccine efficacy, coupled with different levels of implementation effectiveness and background epidemic severity, translate into outcomes. Health Affairs 10\u20131377","DOI":"10.1377\/hlthaff.2020.02054"},{"key":"1592_CR73","doi-asserted-by":"crossref","unstructured":"Pinky L, Dobrovolny HM (2020) SARS-CoV-2 coinfections: Could influenza and the common cold be beneficial? J Med Virol","DOI":"10.1002\/jmv.26098"},{"key":"1592_CR74","doi-asserted-by":"crossref","unstructured":"Plante JA, Liu Y, Liu J, Xia H, Johnson BA, Lokugamage KG, Zhang X, Muruato AE, Zou J, Fontes-Garfias CR et al (2020) Spike mutation D614G alters SARS-CoV-2 fitness. Nature 1\u20136","DOI":"10.1038\/s41586-021-03657-2"},{"key":"1592_CR75","unstructured":"Public Health England: https:\/\/www.gov.uk\/government\/publications\/investigation-of-novel-sars-cov-2-variant-variant-of-concern-20201201 (2021)"},{"key":"1592_CR76","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13054-020-03006-1","volume":"24","author":"P Quah","year":"2020","unstructured":"Quah P, Li A, Phua J (2020) Mortality rates of patients with COVID-19 in the intensive care unit: a systematic review of the emerging literature. Crit Care 24:1\u20134","journal-title":"Crit Care"},{"key":"1592_CR77","doi-asserted-by":"crossref","unstructured":"Rahimi F, Abadi ATB (2021) Implications of the emergence of a new variant of SARS-CoV-2, VUI-202012\/01. Arch Med Res","DOI":"10.1016\/j.arcmed.2021.01.001"},{"key":"1592_CR78","doi-asserted-by":"crossref","unstructured":"Ran L, Chen X, Wang Y, Wu W, Zhang L, Tan X (2020) Risk factors of healthcare workers with corona virus disease 2019: a retrospective cohort study in a designated hospital of Wuhan in China. Clin Infect Dis","DOI":"10.1093\/cid\/ciaa287"},{"key":"1592_CR79","doi-asserted-by":"publisher","first-page":"982","DOI":"10.1016\/j.procs.2021.01.102","volume":"179","author":"ZE Rasjid","year":"2021","unstructured":"Rasjid ZE, Setiawan R, Effendi A (2021) A comparison: Prediction of death and infected COVID-19 cases in Indonesi using time series smoothing and LSTM neural network. Proc Comput Sci 179:982\u2013988","journal-title":"Proc Comput Sci"},{"key":"1592_CR80","doi-asserted-by":"crossref","unstructured":"Reis RF, de Melo Quintela B, de Oliveira Campos J, Gomes JM, Rocha BM, Lobosco M, dos Santos RW (2020) Characterization of the COVID-19 pandemic and the impact of uncertainties, mitigation strategies, and underreporting of cases in South Korea, Italy, and Brazil. Chaos Solit Fract","DOI":"10.1016\/j.chaos.2020.109888"},{"issue":"4","key":"1592_CR81","doi-asserted-by":"publisher","first-page":"584","DOI":"10.1016\/j.jtbi.2009.08.020","volume":"261","author":"D Rios-Doria","year":"2009","unstructured":"Rios-Doria D, Chowell G (2009) Qualitative analysis of the level of cross-protection between epidemic waves of the $$1918-1919$$ influenza pandemic. J Theor Biol 261(4):584\u2013592","journal-title":"J Theor Biol"},{"issue":"2","key":"1592_CR82","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1017\/S0950268899003556","volume":"124","author":"M Roberts","year":"2000","unstructured":"Roberts M, Tobias M (2000) Predicting and preventing measles epidemics in New Zealand: application of a mathematical model. Epidemiol Infect 124(2):279\u2013287","journal-title":"Epidemiol Infect"},{"key":"1592_CR83","doi-asserted-by":"crossref","unstructured":"Saberi M, Hamedmoghadam H, Madani K, Dolk HM, Morgan AS, Morris JK, Khoshnood K, Khoshnood B (2020) Accounting for underreporting in mathematical modeling of transmission and control of COVID-19 in Iran. Front Phys 8","DOI":"10.3389\/fphy.2020.00289"},{"key":"1592_CR84","doi-asserted-by":"crossref","unstructured":"Sarnaglia AJ, Zamprogno B, Molinares FAF, de Godoi LG, Monroy NAJ (2021) Correcting notification delay and forecasting of covid-19 data. J Math Anal Appl p 125202","DOI":"10.1016\/j.jmaa.2021.125202"},{"key":"1592_CR85","doi-asserted-by":"crossref","unstructured":"Shayak B, Sharma MM, Gaur M, Mishra AK (2021) Impact of reproduction number on multiwave spreading dynamics of COVID-19 with temporary immunity: a mathematical model. Int J Infect Dis","DOI":"10.2139\/ssrn.3706045"},{"issue":"10","key":"1592_CR86","doi-asserted-by":"publisher","first-page":"1531","DOI":"10.1038\/s41591-020-1046-6","volume":"26","author":"MK Slifka","year":"2020","unstructured":"Slifka MK, Gao L (2020) Is presymptomatic spread a major contributor to COVID-19 transmission? Nat Med 26(10):1531\u20131533","journal-title":"Nat Med"},{"issue":"2238","key":"1592_CR87","doi-asserted-by":"publisher","first-page":"20200376","DOI":"10.1098\/rspa.2020.0376","volume":"476","author":"RO Stutt","year":"2020","unstructured":"Stutt RO, Retkute R, Bradley M, Gilligan CA, Colvin J (2020) A modelling framework to assess the likely effectiveness of facemasks in combination with lock-down in managing the COVID-19 pandemic. Proc R Soc A 476(2238):20200376","journal-title":"Proc R Soc A"},{"key":"1592_CR88","doi-asserted-by":"publisher","first-page":"109586","DOI":"10.1016\/j.chaos.2019.109586","volume":"132","author":"MA Taneco-Hern\u00e1ndez","year":"2020","unstructured":"Taneco-Hern\u00e1ndez MA, Vargas-De-Le\u00f3n C (2020) Stability and lyapunov functions for systems with Atangana\u2013Baleanu Caputo derivative: an HIV\/AIDS epidemic model. Chaos Solit Fract 132:109586","journal-title":"Chaos Solit Fract"},{"issue":"4","key":"1592_CR89","doi-asserted-by":"publisher","first-page":"368","DOI":"10.1016\/j.bjid.2020.06.003","volume":"24","author":"SC Teixeira","year":"2020","unstructured":"Teixeira SC (2020) Mild and asymptomatic cases of COVID-19 are potential threat for faecal-oral transmission. Br J Infect Dis 24(4):368\u2013368","journal-title":"Br J Infect Dis"},{"key":"1592_CR90","doi-asserted-by":"publisher","unstructured":"Thiagarajan K (2021) Why is India having a covid-19 surge? BMJ 373https:\/\/doi.org\/10.1136\/bmj.n1124. https:\/\/www.bmj.com\/content\/373\/bmj.n1124","DOI":"10.1136\/bmj.n1124"},{"issue":"6","key":"1592_CR91","doi-asserted-by":"publisher","first-page":"1541","DOI":"10.1111\/j.1539-6924.2006.00843.x","volume":"26","author":"KM Thompson","year":"2006","unstructured":"Thompson KM, Duintjer Tebbens RJ, Pallansch MA (2006) Evaluation of response scenarios to potential polio outbreaks using mathematical models. Risk Anal 26(6):1541\u20131556","journal-title":"Risk Anal"},{"key":"1592_CR92","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-78911-6_6","volume-title":"Further notes on the basic reproduction number","author":"P Van den Driessche","year":"2008","unstructured":"Van den Driessche P, Watmough J (2008) Further notes on the basic reproduction number. Springer, New York"},{"key":"1592_CR93","doi-asserted-by":"crossref","unstructured":"van Dorp L, Acman M, Richard D, Shaw LP, Ford CE, Ormond L, Owen CJ, Pang J, Tan CC, Boshier FA, et\u00a0al. (2020) Emergence of genomic diversity and recurrent mutations in SARS-CoV-2. Infect Genet Evolut p. 104351","DOI":"10.1016\/j.meegid.2020.104351"},{"issue":"1","key":"1592_CR94","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1080\/21505594.2021.1877066","volume":"12","author":"C van Oosterhout","year":"2021","unstructured":"van Oosterhout C, Hall N, Ly H, Tyler KM (2021) COVID-19 evolution during the pandemic-implications of new SARS-CoV-2 variants on disease control and public health policies. Virulence 12(1):507","journal-title":"Virulence"},{"key":"1592_CR95","doi-asserted-by":"crossref","unstructured":"Veiga\u00a0e Silva L, de\u00a0Andrade\u00a0Abi MDP, Dos\u00a0Santos AMTB, de\u00a0Mattos\u00a0Teixeira CA, Gomes VHM, Cardoso EHS, da\u00a0Silva MS, Vijaykumar NL, Carvalho SV, Frances CRL, et\u00a0al (2020) COVID-19 mortality underreporting in Brazil: analysis of data from government internet portals. J Med Int Res 22(8), e21413","DOI":"10.2196\/21413"},{"issue":"11","key":"1592_CR96","doi-asserted-by":"publisher","first-page":"1037","DOI":"10.1001\/jama.2021.2294","volume":"325","author":"RP Walensky","year":"2021","unstructured":"Walensky RP, Walke HT, Fauci AS (2021) SARS-CoV-2 variants of concern in the United States-challenges and opportunities. JAMA 325(11):1037\u20131038","journal-title":"JAMA"},{"issue":"10","key":"1592_CR97","doi-asserted-by":"publisher","first-page":"2361","DOI":"10.3201\/eid2610.201702","volume":"26","author":"X Wang","year":"2020","unstructured":"Wang X, Pasco RF, Du Z, Petty M, Fox SJ, Galvani AP, Pignone M, Johnston SC, Meyers LA (2020) Impact of social distancing measures on coronavirus disease healthcare demand, central Texas, USA. Emerg Infect Dis 26(10):2361","journal-title":"Emerg Infect Dis"},{"key":"1592_CR98","doi-asserted-by":"crossref","unstructured":"Wang Y, Wu J, Zhang L, Zhang Y, Wang H, Ding R, Nie J, Li Q, Liu S, Yu Y et al (2021) The infectivity and antigenicity of epidemic SARS-CoV-2 variants in the United Kingdom. Research Square","DOI":"10.21203\/rs.3.rs-153108\/v1"},{"key":"1592_CR99","doi-asserted-by":"publisher","unstructured":"Wise J (2020) Covid-19: New coronavirus variant is identified in UK. BMJ 371. https:\/\/doi.org\/10.1136\/bmj.m4857","DOI":"10.1136\/bmj.m4857"},{"issue":"10234","key":"1592_CR100","doi-asserted-by":"publisher","first-page":"1405","DOI":"10.1016\/S0140-6736(20)30763-7","volume":"395","author":"G Yamey","year":"2020","unstructured":"Yamey G, Sch\u00e4ferhoff M, Hatchett R, Pate M, Zhao F, McDade KK (2020) Ensuring global access to COVID-19 vaccines. The Lancet 395(10234):1405\u20131406","journal-title":"The Lancet"},{"key":"1592_CR101","doi-asserted-by":"crossref","unstructured":"Yan P, Chowell G (2019) Beyond the initial phase: Compartment models for disease transmission. In: Quantitative Methods for investigating infectious disease outbreaks, pp. 135\u2013182. Springer","DOI":"10.1007\/978-3-030-21923-9_5"},{"key":"1592_CR102","first-page":"E021","volume":"54","author":"H Yang","year":"2020","unstructured":"Yang H, Duan G (2020) Analysis on the epidemic factors for the corona virus disease. Zhonghua yu Fang yi xue za zhi [Chinese Journal of Preventive Medicine] 54:E021\u2013E021","journal-title":"Zhonghua yu Fang yi xue za zhi [Chinese Journal of Preventive Medicine]"},{"key":"1592_CR103","doi-asserted-by":"publisher","first-page":"1113","DOI":"10.2105\/AJPH.2021.306220","volume":"111","author":"A Zebrowski","year":"2021","unstructured":"Zebrowski A, Rundle A, Pei S, Yaman T, Yang W, Carr BG, Sims S, Doorley R, Schluger N, Quinn JW et al (2021) A spatiotemporal tool to project hospital critical care capacity and mortality from COVID-19 in US counties. Am J Public Health 111:1113\u20131122","journal-title":"Am J Public Health"},{"issue":"7816","key":"1592_CR104","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1038\/s41586-020-2355-0","volume":"583","author":"X Zhang","year":"2020","unstructured":"Zhang X, Tan Y, Ling Y, Lu G, Liu F, Yi Z, Jia X, Wu M, Shi B, Xu S et al (2020) Viral and host factors related to the clinical outcome of COVID-19. Nature 583(7816):437\u2013440","journal-title":"Nature"},{"key":"1592_CR105","doi-asserted-by":"publisher","unstructured":"Zhang L, Jackson CB, Mou H, Ojha A, Rangarajan ES, Izard T, Farzan M, Choe H (2020) The D614G mutation in the SARS-CoV-2 spike protein reduces S1 shedding and increases infectivity. https:\/\/doi.org\/10.1101\/2020.06.12.148726","DOI":"10.1101\/2020.06.12.148726"},{"key":"1592_CR106","doi-asserted-by":"crossref","unstructured":"Zhao Hj, Lu Xx, Deng Yb, Tang Yj, Lu Jc (2020) COVID-19: asymptomatic carrier transmission is an underestimated problem. Epidemiol Infect pp. 1\u20137","DOI":"10.1017\/S0950268820001235"},{"key":"1592_CR107","doi-asserted-by":"crossref","unstructured":"Zhong L, Mu L, Li J, Wang J, Yin Z, Liu D (2020) Early prediction of the 2019 novel coronavirus outbreak in the mainland China based on simple mathematical model. IEEE Access","DOI":"10.1109\/ACCESS.2020.2979599"},{"key":"1592_CR108","doi-asserted-by":"crossref","unstructured":"Zhu W, Yang J, Lu S, Lan R, Jin D, Luo, X.l., Pu, J., Wu, S., Xu, J. (2020) Beta-and novel Delta-Coronaviruses are identified from wild animals in the qinghai-tibetan plateau, China. Virol Sin pp. 1\u201310","DOI":"10.1007\/s12250-020-00325-z"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01592-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01592-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01592-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,29]],"date-time":"2022-10-29T03:58:22Z","timestamp":1667015902000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01592-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,31]]},"references-count":108,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1592"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01592-6","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,31]]},"assertion":[{"value":"18 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 July 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}],"article-number":"199"}}